| PREFACE | | xiii | |
| 1 Kevin Bacon, the Small World, and Why It All Matters |
| | 3 | (6) |
| PART I STRUCTURE | | 9 | (154) |
| 2 An Overview of the Small-World Phenomenon |
| | 11 | (30) |
| 2.1 Social Networks and the Small World |
| | 11 | (14) |
| 2.1.1 A Brief History of the Small World |
| | 12 | (8) |
| 2.1.2 Difficulties with the Real World |
| | 20 | (4) |
| 2.1.3 Reframing the Question to Consider All Worlds |
| | 24 | (1) |
| 2.2 Background on the Theory of Graphs |
| | 25 | (16) |
| | 25 | (2) |
| 2.2.2 Length and Length Scaling |
| | 27 | (4) |
| 2.2.3 Neighbourhoods and Distribution Sequences |
| | 31 | (1) |
| | 32 | (1) |
| 2.2.5 "Lattice Graphs" and Random Graphs |
| | 33 | (6) |
| 2.2.6 Dimension and Embedding of Graphs |
| | 39 | (2) |
| 3 Big Worlds and Small Worlds: Models of Graphs |
| | 41 | (60) |
| | 42 | (49) |
| | 42 | (24) |
| 3.1.2 A Stripped-Down Model: Beta-Graphs |
| | 66 | (4) |
| 3.1.3 Shortcuts and Contractions: Model Invariance |
| | 70 | (17) |
| 3.1.4 Lies, Damned Lies, and (More) Statistics |
| | 87 | (4) |
| | 91 | (9) |
| 3.2.1 Uniform Spatial Graphs |
| | 93 | (5) |
| 3.2.2 Gaussian Spatial Graphs |
| | 98 | (2) |
| 3.3 Main Points in Review |
| | 100 | (1) |
| 4 Explanations and Ruminations |
| | 101 | (37) |
| | 101 | (13) |
| 4.1.1 The Connected-Caveman World |
| | 102 | (7) |
| 4.1.2 Moore Graphs as Approximate Random Graphs |
| | 109 | (5) |
| 4.2 Transitions in Relational Graphs |
| | 114 | (13) |
| 4.2.1 Local and Global Length Scales |
| | 114 | (2) |
| 4.2.2 Length and Length Scaling |
| | 116 | (1) |
| 4.2.3 Clustering Coefficient |
| | 117 | (1) |
| | 118 | (2) |
| 4.2.5 Results and Comparisons with Beta-Model |
| | 120 | (7) |
| 4.3 Transitions in Spatial Graphs |
| | 127 | (6) |
| 4.3.1 Spatial Length versus Graph Length |
| | 127 | (1) |
| 4.3.2 Length and Length Scaling |
| | 128 | (2) |
| | 130 | (2) |
| 4.3.4 Results and Comparisons |
| | 132 | (1) |
| 4.4 Variations on Spatial and Relational Graphs |
| | 133 | (3) |
| 4.5 Main Points in Review |
| | 136 | (2) |
| 5 "It's a Small World after All": Three Real Graphs |
| | 138 | (25) |
| | 140 | (7) |
| 5.1.1 Examining the Graph |
| | 141 | (2) |
| | 143 | (4) |
| 5.2 The Power of Networks |
| | 147 | (6) |
| 5.2.1 Examining the System |
| | 147 | (3) |
| | 150 | (3) |
| | 153 | (6) |
| 5.3.1 Examining the System |
| | 154 | (2) |
| | 156 | (3) |
| | 159 | (2) |
| 5.5 Main Points in Review |
| | 161 | (2) |
| PART II DYNAMICS | | 163 | (80) |
| 6 The Spread of Infectious Disease in Structured Populations |
| | 165 | (16) |
| 6.1 A Brief Review of Disease Spreading |
| | 166 | (2) |
| | 168 | (12) |
| 6.2.1 Introduction of the Problem |
| | 168 | (1) |
| 6.2.2 Permanent-Removal Dynamics |
| | 169 | (7) |
| 6.2.3 Temporary-Removal Dynamics |
| | 176 | (4) |
| 6.3 Main Points in Review |
| | 180 | (1) |
| 7 Global Computation in Cellular Automata |
| | 181 | (18) |
| | 181 | (6) |
| | 184 | (3) |
| 7.2 Cellular Automata on Graphs |
| | 187 | (11) |
| 7.2.1 Density Classification |
| | 187 | (8) |
| | 195 | (3) |
| 7.3 Main Points in Review |
| | 198 | (1) |
| 8 Cooperation in a Small World: Games on Graphs |
| | 199 | (24) |
| | 199 | (9) |
| 8.1.1 The Prisoner's Dilemma |
| | 200 | (4) |
| 8.1.2 Spatial Prisoner's Dilemma |
| | 204 | (2) |
| 8.1.3 N-Player Prisoner's Dilemma |
| | 206 | (1) |
| 8.1.4 Evolution of Strategies |
| | 207 | (1) |
| 8.2 Emergence of Cooperation in a Homogeneous Population |
| | 208 | (11) |
| 8.2.1 Generalised Tit-for-Tat |
| | 209 | (5) |
| 8.2.2 Win-Stay, Lose-Shift |
| | 214 | (5) |
| 8.3 Evolution of Cooperation in a Heterogeneous Population |
| | 219 | (2) |
| 8.4 Main Points in Review |
| | 221 | (2) |
| 9 Global Synchrony in Populations of Coupled Phase Oscillators |
| | 223 | (17) |
| | 223 | (5) |
| 9.2 Kuramoto Oscillators on Graphs |
| | 228 | (10) |
| 9.3 Main Points in Review |
| | 238 | (2) |
| | 240 | (3) |
| NOTES | | 243 | (6) |
| BIBLIOGRAPHY | | 249 | (8) |
| INDEX | | 257 | |
